Manuscript Em03-G@1 · Em03-G-v1.tex · section · 0bb3a8a3

Coefficients and Weyl normalization

Field conventions and the independent background › Coefficients and Weyl normalization · lines 369–425

Em03-G@1#eq:coeff

Use horizon-incident and infinity-incident modes $q_{H}$ , $q_{I}$ , normalized as in Em03-D–F. Their spectral transform is

(U u)_{j} (ω) = \frac{1}{2 π} \int q_{j} (x, ω) u (x) d x .

It is unitary from $L^{2} (R)$ onto $L^{2} ((0, \infty); C^{2})$ , and takes $L$ to multiplication by $ω^{2}$ .

The field expansion is

Ψ (t, x) = j \sum \int_{0}^{\infty} \frac{q _{j} a _{j} e ^{- i ω t} + q _{j} a _{j} ^{†} e ^{i ω t}}{4 π ω} d ω .

Consequently

k_{j} (f) (ω) = \frac{1}{4 π ω} \int f (t, x) q_{j} (x, ω) e^{i ω t} d t d x .

Annihilation tests are conjugate-linear:

a (k) = j \sum \int k_{j} (ω) a_{j} (ω) d ω .

Set

X (k) = a (k) + a (k)^{†}, W (k) = e^{i X (k)},

s (k, l) = 2 Im ⟨ k, l ⟩, W (k) W (l) = e^{- i s (k, l) /2} W (k + l) .

Then $ϕ (F [f]) = X (k (f))$ .

Define

C = diag (1 + 2 n_{H}, 1), ∥ k ∥_{U}^{2} = ⟨ k, C k ⟩, q (k) = e^{- ∥ k ∥_{U}^{2} /2} .

The ordinary two-channel norm is $∥ \cdot ∥_{0}$ . We have

ω_{U} (W (k)) = q (k), ω_{U} (X (k)^{2}) = ∥ k ∥_{U}^{2}, ∥ k ∥_{0} \leq ∥ k ∥_{U} .

Displayed formulas in this passage

Em03-G@1#eq:coeff · line 391
$k_{j} (f) (ω) = \frac{1}{4 π ω} \int f (t, x) q_{j} (x, ω) e^{i ω t} d t d x .$
unlabeled · line 374
$(U u)_{j} (ω) = \frac{1}{2 π} \int q_{j} (x, ω) u (x) d x .$
unlabeled · line 384
$Ψ (t, x) = j \sum \int_{0}^{\infty} \frac{q _{j} a _{j} e ^{- i ω t} + q _{j} a _{j} ^{†} e ^{i ω t}}{4 π ω} d ω .$
unlabeled · line 399
$a (k) = j \sum \int k_{j} (ω) a_{j} (ω) d ω .$
unlabeled · line 403
$X (k) = a (k) + a (k)^{†}, W (k) = e^{i X (k)},$
unlabeled · line 406
$s (k, l) = 2 Im ⟨ k, l ⟩, W (k) W (l) = e^{- i s (k, l) /2} W (k + l) .$
unlabeled · line 413
$C = diag (1 + 2 n_{H}, 1), ∥ k ∥_{U}^{2} = ⟨ k, C k ⟩, q (k) = e^{- ∥ k ∥_{U}^{2} /2} .$
unlabeled · line 420
$ω_{U} (W (k)) = q (k), ω_{U} (X (k)^{2}) = ∥ k ∥_{U}^{2}, ∥ k ∥_{0} \leq ∥ k ∥_{U} .$